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Properties

Label 960.3.x.e
Level $960$
Weight $3$
Character orbit 960.x
Analytic conductor $26.158$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(97,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.x (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{17} - 144 q^{19} - 168 q^{25} + 192 q^{33} + 144 q^{35} + 192 q^{41} + 192 q^{43} - 864 q^{59} - 456 q^{65} - 336 q^{67} + 792 q^{73} - 48 q^{75} - 216 q^{81} + 240 q^{83} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −1.22474 − 1.22474i 0 −2.27993 + 4.44994i 0 0.340581 + 0.340581i 0 3.00000i 0
97.2 0 −1.22474 − 1.22474i 0 2.27993 − 4.44994i 0 −0.340581 − 0.340581i 0 3.00000i 0
97.3 0 −1.22474 − 1.22474i 0 −4.99965 − 0.0587820i 0 2.54785 + 2.54785i 0 3.00000i 0
97.4 0 −1.22474 − 1.22474i 0 4.99965 + 0.0587820i 0 −2.54785 − 2.54785i 0 3.00000i 0
97.5 0 −1.22474 − 1.22474i 0 −1.30551 − 4.82656i 0 8.64266 + 8.64266i 0 3.00000i 0
97.6 0 −1.22474 − 1.22474i 0 1.30551 + 4.82656i 0 −8.64266 − 8.64266i 0 3.00000i 0
97.7 0 1.22474 + 1.22474i 0 −2.94848 − 4.03813i 0 1.01191 + 1.01191i 0 3.00000i 0
97.8 0 1.22474 + 1.22474i 0 2.94848 + 4.03813i 0 −1.01191 − 1.01191i 0 3.00000i 0
97.9 0 1.22474 + 1.22474i 0 −3.24004 + 3.80817i 0 5.40656 + 5.40656i 0 3.00000i 0
97.10 0 1.22474 + 1.22474i 0 3.24004 − 3.80817i 0 −5.40656 − 5.40656i 0 3.00000i 0
97.11 0 1.22474 + 1.22474i 0 −1.70578 + 4.70004i 0 −8.96895 − 8.96895i 0 3.00000i 0
97.12 0 1.22474 + 1.22474i 0 1.70578 − 4.70004i 0 8.96895 + 8.96895i 0 3.00000i 0
673.1 0 −1.22474 + 1.22474i 0 −2.27993 − 4.44994i 0 0.340581 − 0.340581i 0 − 3.00000i 0
673.2 0 −1.22474 + 1.22474i 0 2.27993 + 4.44994i 0 −0.340581 + 0.340581i 0 − 3.00000i 0
673.3 0 −1.22474 + 1.22474i 0 −4.99965 + 0.0587820i 0 2.54785 − 2.54785i 0 − 3.00000i 0
673.4 0 −1.22474 + 1.22474i 0 4.99965 − 0.0587820i 0 −2.54785 + 2.54785i 0 − 3.00000i 0
673.5 0 −1.22474 + 1.22474i 0 −1.30551 + 4.82656i 0 8.64266 − 8.64266i 0 − 3.00000i 0
673.6 0 −1.22474 + 1.22474i 0 1.30551 − 4.82656i 0 −8.64266 + 8.64266i 0 − 3.00000i 0
673.7 0 1.22474 − 1.22474i 0 −2.94848 + 4.03813i 0 1.01191 − 1.01191i 0 − 3.00000i 0
673.8 0 1.22474 − 1.22474i 0 2.94848 − 4.03813i 0 −1.01191 + 1.01191i 0 − 3.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.e even 4 1 inner
40.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.x.e ✓ 24
4.b odd 2 1 960.3.x.f yes 24
5.c odd 4 1 960.3.x.f yes 24
8.b even 2 1 960.3.x.f yes 24
8.d odd 2 1 inner 960.3.x.e ✓ 24
20.e even 4 1 inner 960.3.x.e ✓ 24
40.i odd 4 1 inner 960.3.x.e ✓ 24
40.k even 4 1 960.3.x.f yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.3.x.e ✓ 24 1.a even 1 1 trivial
960.3.x.e ✓ 24 8.d odd 2 1 inner
960.3.x.e ✓ 24 20.e even 4 1 inner
960.3.x.e ✓ 24 40.i odd 4 1 inner
960.3.x.f yes 24 4.b odd 2 1
960.3.x.f yes 24 5.c odd 4 1
960.3.x.f yes 24 8.b even 2 1
960.3.x.f yes 24 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{24} + 51792 T_{7}^{20} + 751327584 T_{7}^{16} + 2102660685056 T_{7}^{12} + 341710471590144 T_{7}^{8} + \cdots + 75119246442496 \) Copy content Toggle raw display
\( T_{19}^{6} + 36T_{19}^{5} - 252T_{19}^{4} - 20416T_{19}^{3} - 178656T_{19}^{2} + 43008T_{19} + 74752 \) Copy content Toggle raw display